Báo cáo toán học: "A Gessel–Viennot-Type Method for Cycle Systems in a Directed Graph" doc

A Gessel–Viennot-Type Method for Cycle Systems
in a Directed Graph
Christopher R. H. Hanusa
Department of Mathematical Sciences
Binghamton University, Binghamton, New York, USA

Submitted: Nov 28, 2005; Accepted: Mar 31, 2006; Published: Apr 4, 2006
Mathematics Subject Classifications: Primary 05B45, 05C30;
Secondary 05A15, 05B20, 05C38, 05C50, 05C70, 11A51, 11B83, 15A15, 15A36, 52C20
Keywords: directed graph, cycle system, path system, walk system, Aztec diamond,
Aztec pillow, Hamburger Theorem, Kasteleyn–Percus, Gessel–Viennot, Schr¨oder numbers
Abstract
We introduce a new determinantal method to count cycle systems in a directed
graph that generalizes Gessel and Viennot’s determinantal method on path systems.
The method gives new insight into the enumeration of domino tilings of Aztec
diamonds, Aztec pillows, and related regions.
1 Introduction
In this article, we present an analogue of the Gessel–Viennot method for counting cycle
systems on a type of directed graph we call a hamburger graph. A hamburger graph H is
made up of two acyclic graphs G
1
and G
2
and a connecting edge set E
3
with the following
properties. The graph G
1
has k distinguished vertices {v
1
, ,v

k
} with directed paths
from v
i
to v
j
only if i<j. The graph G
2
has k distinguished vertices {w
k+1
, ,w
2k
} with
directed paths from w
i
to w
j
only if i>j.TheedgesetE
3
connects the vertices v
i
and
w
k+i
by way of edges e
i
: v
i
→ w
k+i

and e

i
: w
k+i
→ v
i
. (See Figure 1 for a visualization.)
Hamburger graphs arise naturally in the study of Aztec diamonds, as explained in Section
5.
The Gessel–Viennot method is a determinantal method to count path systems in an
acyclic directed graph G with k sources s
1
, ,s
k
and k sinks t
1
, ,t
k
.Apath system
P is a collection of k vertex-disjoint paths, each one directed from s
i
to t
σ(i)
, for some
permutation σ ∈ S
k
(where S
k
is the symmetric group on k elements). Call a path

system P positive if the sign of this permutation σ satisfies sgn(σ)=+1andnegative if
sgn(σ)=−1. Let p
+
be the number of positive path systems and p
−
be the number of
negative path systems.
the electronic journal of combinatorics 13 (2006), #R37 1
v
k
w
k+1
w
k+2
w
2k
G
1
E
3
G
2
e

1
e
1
v
2
v

1
Figure 1: A hamburger graph
Corresponding to this graph G is a k × k matrix A =(a
ij
), where a
ij
is the number of
paths from s
i
to t
j
in G. The result of Gessel and Viennot states that det A = p
+
− p
−
.
The Gessel–Viennot method was introduced in [4, 5], and has its roots in works by Karlin
and McGregor [8] and Lindstr¨om [10]. A nice exposition of the method and applications
is given in the article by Aigner [1].
This article concerns a similar determinantal method for counting cycle systems in a
hamburger graph H.Acycle system C is a collection of vertex-disjoint directed cycles in
H.Letl be the number of edges in C that travel from G
2
to G
1
and let m be the number
of cycles in C. Call a cycle system positive if (−1)
l+m
=+1andnegative if (−1)
l+m

= −1.
Let c
+
be the number of positive cycle systems and c
−
be the number of negative cycle
systems. Corresponding to each hamburger graph H is a 2k × 2k block matrix M
H
of the
form
M
H
=

AI
k
−I
k
B

,
where in the upper triangular matrix A =(a
ij
), a
ij
is the number of paths from v
i
to v
j
in G

1
and in the lower triangular matrix B =(b
ij
), b
ij
is the number of paths from w
k+i
to w
k+j
in G
2
. This matrix M
H
is referred to as a hamburger matrix.
Theorem 1.1 (The Hamburger Theorem). If H is a hamburger graph, then det M
H
=
c
+
− c
−
.
A hamburger graph H is called strongly planar if there is a planar embedding of H
that sends v
i
to (i, 1) and w
k+i
to (i, −1) for all 1 ≤ i ≤ k, and keeps edges of E
1
in the

half-space y ≥ 1andedgesofE
2
in the half-space y ≤−1. This definition suggests that
G
1
and G
2
are “relatively” planar in H, a stronger condition than planarity of H.Notice
that when H is strongly planar, each cycle must use exactly one edge from G
2
to G
1
.
Hence, the sign of every cycle system is +1. This implies the following corollary.
Corollary 1.2. If H is a strongly planar hamburger graph, det M
H
= c
+
.
The following simple example serves to guide us. Consider the two graphs G
1
=
(V
1
,E
1
)andG
2
=(V
2

,E
2
), where V
1
= {v
1
,v
2
,v
3
,v}, V
2
= {w
4
,w
5
,w
6
,w}, E
1
= {v
1
→
v
2
,v
2
→ v
3
,v

1
→ v, v → v
3
},andE
2
= {w
6
→ w
5
,w
5
→ w
4
,w
6
→ w, w → w
4
}.Our
the electronic journal of combinatorics 13 (2006), #R37 2
v
∗
v
1
v
3
w
4
w
6
v

2
w
∗
w
5
Figure 2: A simple hamburger graph H
hamburger graph H will be the union of G
1
, G
2
, and the edge set E
3
.Inthisexample,
k =3andH is strongly planar. Figure 2 gives a graphical representation of H.
In this example, the hamburger matrix M
H
equals
M
H
=








1 1 2 100
0 1 1 010

0 0 1 001
−1 0 0 100
0 −1 0 110
00−1211








.
The determinant of M
H
is 17, corresponding to the seventeen cycle systems (each with
sign +1) in Figure 3.
The graph that inspired the definition of a hamburger graph comes from the work
of Brualdi and Kirkland [2], in which they give a new proof that the number of domino
tilings of the Aztec diamond is 2
n(n+1)/2
.AnAztec diamond, denoted by AD
n
,isthe
union of the 2n(n+1) unit squares with integral vertices (x, y) such that |x| + |y|≤n +1.
See Figure 4 for an example of an Aztec diamond, as well as an example of an Aztec
pillow and a generalized Aztec pillow, described in the next paragraphs.
An Aztec pillow, as it was initially presented in [12], is also a rotationally symmetric
region in the plane. On the top left boundary, however, the steps are composed of three
squares to the right for every square up. Another definition is that Aztec pillows are

the union of the unit squares with integral vertices (x, y) such that |x + y| <n+1and
|3y − x| <n+ 3. As with Aztec diamonds, we denote the Aztec pillow with 2n squares
in each of the central rows by AP
n
. In Section 6, we extend the notion of Aztec pillows
having steps of length 3 to “odd pillows”—those that have steps that are of a constant
odd length. The integral vertices (x, y) of the unit squares in q-pillows for q odd satisfy
|x + y| <n+1and|qy − x| <n+ q.
We introduce the idea of a generalized Aztec pillow, where the steps on all diagonals
are of possibly different odd lengths. More specifically, a generalized Aztec pillow is a
horizontally convex and vertically convex region such that the steps both up and down in
each diagonal have an odd number of squares horizontally for every one square vertically.
the electronic journal of combinatorics 13 (2006), #R37 3
Figure 3: The seventeen cycle systems for the hamburger graph in Figure 2
Figure 4: Examples of an Aztec diamond, an Aztec pillow, and a generalized Aztec pillow
the electronic journal of combinatorics 13 (2006), #R37 4
A key fact that we will use is that any generalized Aztec pillow can be recovered from a
large enough Aztec diamond by the placement of horizontal dominoes.
Brualdi and Kirkland prove the formula for the number of domino tilings of an Aztec
diamond by creating an associated digraph and counting its cycle systems, manipulating
the digraph’s associated Kasteleyn–Percus matrix of order n(n +1). To learn about
Kasteleyn theory and Kasteleyn–Percus matrices, start with Kasteleyn’s 1961 work [9]
and Percus’s 1963 work [11]. The Hamburger Theorem proves that we can count the
number of domino tilings of an Aztec diamond with a much smaller determinant, of order
2n. A Schur complement allows us to reduce the determinant calculation to one of order
n. An analogous reduction in determinant size (from order O(n
2
)toorderO(n)) occurs
for all regions to which this theorem applies, including generalized Aztec pillows. In
addition, whereas Kasteleyn theory applies only to planar graphs, there is no planarity

restriction for hamburger graphs. For this reason, the Hamburger Theorem gives a new
counting method for cycle systems in some non-planar graphs.
More recently, Eu and Fu present a new proof of the number of tilings of an Aztec
diamond [3]. Their lattice-path-based proof also reduces to an n×n determinant but does
not generalize to the case of Aztec pillows. This result is discussed further in Section 5.2.
In Section 2, we present an overview of the proof of the Hamburger Theorem, including
the key lemmas involved. The necessary machinery is built up in Section 3 to complete
the proof in Section 4. Section 5 presents applications of the Hamburger Theorem to
Aztec diamonds, Aztec pillows, and generalized Aztec pillows. Section 6 concludes with
a counterexample to the most natural generalization of the Hamburger Theorem and an
extension of Propp’s Conjecture on Aztec pillows.
2 Outline of the Proof of the Hamburger Theorem
2.1 The Hamburger Theorem
Like the proof of the Gessel–Viennot method, the proof of the Hamburger Theorem hinges
on cancellation of terms in the permutation expansion of the determinant of M
H
.Inthe
proof, we must allow closed directed walks in addition to cycles. We must also allow walk
systems, arbitrary collections of closed directed walks, since they can and will appear
in the permutation expansion of the hamburger determinant. We call a walk system
simple if the set of walks visits no vertex more than once. We call a cycle of the form
c : v
i
→ w
i+k
→ v
i
a 2-cycle.
Each signed term in the permutation expansion of the hamburger determinant is the
contribution of many signed walk systems W. Walk systems that are not cycle systems

will all cancel out in the determinant expansion. We will show this in two steps. We
start by considering walk systems that are not simple. If this is the case, one of the two
following properties MAY hold.
Property 1. The walk system contains a walk that has a self-intersection.
Property 2. The walk system has two intersecting walks, neither of which is a 2-cycle.
the electronic journal of combinatorics 13 (2006), #R37 5
The following lemma shows that the contributions of walk systems satisfying either of
these two properties cancel in the permutation expansion of the determinant of M
H
.
Lemma 2.1. The set of all walk systems W that satisfy either Property 1 or Property
2 can be partitioned into equivalence classes, each of which contributes a net zero to the
permutation expansion of the determinant of M
H
.
The proof of Lemma 2.1 uses a generalized involution principle. Walk systems cancel
in families based on the their “first” intersection point.
The remainder of the cancellation in the determinant expansion is based on the concept
of a minimal walk system; we motivate this definition by asking the following questions.
What kind of walk systems does the permutation expansion of the hamburger determinant
generate, and how is this different from our original notion of cycle systems that we wanted
to count in the introduction? The key difference is that the same collection of walks can
be generated by multiple terms in the determinantal expansion of M
H
; whereas, we would
only want to count it once as a cycle system. This redundancy arises when the walk visits
three distinguished vertices in G
1
without passing via G
2

or vice versa. We illustrate this
notion with the following example.
Consider the second cycle system in the third row of Figure 3, consisting of one
solitary directed cycle. Since this cycle visits vertices v
1
, v
2
, v
3
, w
6
,andw
4
in that
order, it contributes a non-zero weight in the permutation expansion of the determinant
corresponding to the term (12364) in S
6
. Notice that this cycle also contributes a non-
zero weight in the permutation expansion of the determinant corresponding to the term
(1364). We see this since our cycle follows a path from v
1
to v
3
(by way of v
2
), returning
to v
1
via w
6

and w
4
. We must deal with this ambiguity. We introduce the idea of a
minimal permutation cycle, one which does not include more than two successive entries
with values between 1 and k or between k +1 and2k. We see that (1364) is minimal
while (12364) is not.
We notice that walk systems arise from permutations, so it is natural to think of
a walk system as a permutation together with a collection of walks that “follow” the
permutation. This is the idea of a walk system–permutation pair (or WSP-pair for short)
that is presented in Section 3.4. From the idea of a minimal permutation cycle, we define
a minimal walk to have as its base permutation a minimal permutation cycle, and a
minimal walk system to be composed of only minimal walks. Since our original goal was
to count “cycle systems” in a directed graph, we realize we need to be precise and instead
count “simple minimal walk systems”. This leads to the second part of the proof of the
Hamburger Theorem.
Given a walk system that is either not simple or not minimal and that satisfies neither
Property 1 nor Property 2, at least one of the two following properties MUST hold.
Property 3. The walk system has two intersecting walks, one of which is a 2-cycle.
Property 4. The walk system is not minimal.
The following lemma shows that the contributions of walk systems satisfying either of
these new properties cancel in the permutation expansion of the determinant of M
H
.
the electronic journal of combinatorics 13 (2006), #R37 6
Lemma 2.2. The set of all walk systems W that satisfy neither Property 1 nor Property
2 and that satisfy Property 3 or Property 4 can be partitioned into equivalence classes,
each of which contributes a net zero to the permutation expansion of the determinant of
M
H
.

The proof of Lemma 2.2 is also based on involutions. Walk systems cancel in families
built from an index set containing the set of all 2-cycle intersections and non-minimalities.
If a walk system satisfies none of the conditions of Properties 1 through 4, then it is
indeed a simple minimal walk system, or in other words, a cycle system. The cancellation
from the above sets of families gives that only cycle systems contribute to the permutation
expansion of the determinant of M
H
. This contribution is the signed weight of each cycle
system, so the determinant of M
H
exactly equals c
+
− c
−
. Theorem 1.1 follows from
Lemmas 2.1 and 2.2 in Section 4. •
2.2 The Weighted Hamburger Theorem
There is also a weighted version of the Hamburger Theorem, and it will be under this
generalization that Lemmas 2.1 and 2.2 are proved. We allow weights wt(e)ontheedgesof
the hamburger graph; the simplest weighting, which counts the number of cycle systems,
assigns wt(e) ≡ 1. We require that wt(e
i
)wt(e

i
) = 1 for all 2 ≤ i ≤ k − 1, but we do
not require this condition for i = 1 nor for i = k. Define the 2k × 2k weighted hamburger
matrix M
H
to be the block matrix

M
H
=

AD
1
−D
2
B

. (1)
In the upper-triangular k × k matrix A =(a
ij
), a
ij
is the sum of the products of the
weights of edges over all paths from v
i
to v
j
in G
1
. In the lower triangular k × k matrix
B =(b
ij
), b
ij
is the sum of the products of the weights of edges over all paths from w
k+i
to

w
k+j
in G
2
. The diagonal k × k matrix D
1
has as its entries d
ii
=wt(e
i
) and the diagonal
k × k matrix D
2
has as its entries d
ii
=wt(e

i
). Note that when the weights of the edges
in E
3
are all 1, these matrices satisfy D
1
= D
2
= I
k
.
We wish to count vertex-disjoint unions of weighted cycles in H. In any hamburger
graph H, there are two possible types of cycle. There are k 2-cycles

c : v
i
e
i
−→ w
k+i
e

i
−→ v
i
and many more general cycles that alternate between G
1
and G
2
.Wecanthinkofa
general cycle as a path P
1
in G
1
connected by an edge e
1,1
∈ E
3
to a path Q
1
in G
2
,
which in turn connects to a path P

2
in G
1
by an edge e

1,2
, continuing in this fashion until
arriving at a final path Q
l
in G
2
whose terminal vertex is adjacent to the initial vertex of
P
1
. We write
c : P
1
e
1,1
−→ Q
1
e

1,2
−→ P
2
e
2,1
−→···
e

l,1
−→ P
l
e

l,2
−→ Q
l
.
the electronic journal of combinatorics 13 (2006), #R37 7
For each cycle c, we define the weight wt(c)ofc to be the product of the weights of all
edges traversed by c:
wt(c)=

e∈c
wt(e).
We define a weighted cycle system to be a collection C of m vertex-disjoint cycles. We
again define the sign of a weighted cycle system to be sgn(C)=(−1)
l+m
,wherel is the
total number of edges from G
2
to G
1
in C. We say that a weighted cycle system C is
positive if sgn(C)=+1andnegative if sgn(C)=−1. For a hamburger graph H,letc
+
be
the sum of the weights of positive weighted cycle systems, and let c
−

be the sum of the
weights of negative weighted cycle systems.
Theorem 2.3 (The weighted Hamburger Theorem). The determinant of the weigh-
ted hamburger matrix M
H
equals c
+
− c
−
.
As above, Theorem 2.3 follows from Lemmas 2.1 and 2.2. The proofs will be presented
after developing the following necessary machinery.
3 Additional Definitions
3.1 Edge Cycles and Permutation Cycles
In the proof of the Hamburger Theorem, there are two distinct mathematical objects that
have the name “cycle”. We have already mentioned the type of cycle that appears in
graph theory. There, a (simple) cycle in a directed graph is a closed directed path with
no repeated vertices.
Secondly, there is a notion of cycle when we talk about permutations. If σ ∈ S
n
is a
permutation, we can write σ as the product of disjoint cycles σ = χ
1
χ
2
···χ
τ
.
To distinguish between these two types of cycles when confusion is possible, we call the
former kind an edge cycle and the latter kind a permutation cycle. Notationally, we use

Roman letters when discussing edge cycles and Greek letters when discussing permutation
cycles.
3.2 Permutation Expansion of the Determinant
We recall that the permutation expansion of the determinant of an n×n matrix M =(m
ij
)
is the expansion of the determinant as
det M =

σ∈S
n
(sgn σ)m
1,σ(1)
···m
n,σ(n)
. (2)
We will be considering non-zero terms in the permutation expansion of the determinant
of the hamburger matrix M
H
. Because of the special block form of the hamburger matrix
in Equation (1), the permutations σ that make non-zero contributions to this sum are
products of disjoint cycles of either of two forms—the simple transposition
χ =(ϕ
11
ω
11
)
the electronic journal of combinatorics 13 (2006), #R37 8
or the general permutation cycle
χ =(ϕ

11
ϕ
12
··· ϕ
1µ
1
ω
11
ω
12
··· ω
1ν
1
ϕ
21
······ ϕ
λµ
λ
ω
λ1
··· ω
λν
λ
). (3)
In the first case, ω
11
= ϕ
11
+ k. In the second case, 1 ≤ ϕ
ικ

≤ k, k +1 ≤ ω
ικ
≤ 2k,
ϕ
ικ
<ϕ
ι,κ+1
,andω
ικ
>ω
ι,κ+1
for all 1 ≤ ι ≤ λ and relevant κ. The block matrix form also
implies that ϕ
ιµ
ι
+ k = ω
ι1
, ω
ιν
ι
− k = ϕ
ι+1,1
,andω
λν
λ
− k = ϕ
11
. These last requirements
along with the fact that no integers appear more than once in a permutation cycle imply
that µ

i
,ν
i
≥ 2 for 1 ≤ i ≤ λ. So that this permutation cycle is in standard form, we make
sure that ϕ
11
=min
ι,κ
ϕ
ικ
. In order to refer to this value later, we define a function Φ by
Φ(χ)=ϕ
11
. Each value 1 ≤ ϕ
ι
≤ k or k +1≤ ω
ι
≤ 2k appears at most once for any
σ ∈ S
2k
.
We call a permutation cycle χ minimal if it is a transposition or if µ
ι
= ν
ι
= 2 for all
ι. Minimality implies that we can write our general permutation cycles χ in the form
χ =(ϕ
11
ϕ

12
ω
11
ω
12
ϕ
21
··· ϕ
λ2
ω
λ1
ω
λ2
), (4)
with the same conditions as before. We call a permutation σ = χ
1
···χ
τ
minimal if each
of its cycles χ
ι
is minimal.
3.3 Walks Associated to a Permutation
To each permutation cycle χ ∈ S
2k
, we can associate one or more walks c
χ
in H.
If χ is the transposition χ =(ϕ
11

ω
11
), then we associate the 2-cycle c
χ
: v
ϕ
11
→
w
ω
11
→ v
ϕ
11
to χ. To any permutation cycle χ that is not a transposition, we can
associate multiple walks c
χ
by gluing together paths that follow χ in the following way.
If χ has the form of Equation (3), then for each 1 ≤ i ≤ λ,letP
i
be any path in G
1
that
visits each of the vertices v
ϕ
i1
, v
ϕ
i2
, all the way through v

ϕ
iµ
i
in order. Similarly, let Q
i
be any path in G
2
that visits each of the vertices w
ω
i1
, w
ω
i2
, through w
ω
iν
i
in order. For
each choice of paths P
i
and Q
i
, we have a possibility for the walk c
χ
;wecanset
c
χ
: P
1
e

ϕ
12
−→ Q
1
e

ϕ
21
−→ P
2
e
ϕ
22
−→······
e

ϕ
λ1
−→ P
λ
e
ϕ
λ2
−→ Q
λ
. (5)
See Figure 5 for the choices of c
(12364)
in the hamburger graph presented in Figure 2.
We call λ the number of P -paths in c

χ
. The function Φ, defined in the previous section,
defines a partial ordering on walks in a walk system—we say that the associated walk c
χ
comes before the associated walk c
χ

if Φ(χ) < Φ(χ

). We call this the initial term order.
As in Section 2.2, we define the weight of a walk c
χ
to be the product of the weights of
all edges traversed by c
χ
.
3.4 Walk System-Permutation Pairs
We defined walk systems in Section 2, but we will see that the proof of Theorem 1.1
requires us to think of walk systems first as a permutation and second as a collection of
the electronic journal of combinatorics 13 (2006), #R37 9
orχ =(12364)
Figure 5: A permutation cycle χ and the two walks in H associated to χ
walks determined by the permutation. We will see that for cycle systems as presented
initially, signs and weights are not changed by this recharacterization.
If H is a hamburger graph with k pairs of distinguished vertices, we define a walk
system–permutation pair as follows.
Definition 3.1. A walk system–permutation pair (or WSP-pair for short) is a pair (W,σ),
where σ ∈ S
2k
is a permutation and W is a collection of walks c ∈W with the following

property: if the disjoint cycle representation of σ is σ = χ
1
···χ
τ
,thenW is a collection
of τ walks c
χ
ι
, for 1 ≤ ι ≤ τ,wherec
χ
ι
is a walk associated to the permutation cycle χ
ι
.
We define the weight of a WSP-pair (W,σ) to be the product of the weights of the
associated walks c
χ
∈W.
Each permutation σ yields many collections of walks W, collections of walks W may
be associated to many permutations σ, but any walk system W corresponds to one and
only one minimal permutation σ
m
. This is because, given any path as in Equation (5),
we can read off the initial and terminal vertices of each P
i
and Q
i
in order, producing a
well-defined permutation cycle σ
m

. We define a WSP-pair (W,σ)tobeminimal if σ is a
minimal permutation.
ForaWSP-pair(W,σ), where σ = χ
1
···χ
τ
, we define the sign of the WSP-pair,
sgn(W,σ), to be (−1)
l
sgn(σ), where λ
χ
is the number of P -paths in c
χ
and where l =

c
χ
∈W
λ
χ
. Alternatively, we could consider the sign of (W,σ) to be the product of the
signs of its associated walks c
χ
, where the sign of c
χ
is sgn(c
χ
)=(−1)
λ
χ

sgn(χ). We say
that a WSP-pair (W,σ)ispositive if sgn(W,σ)=+1andisnegative if sgn(W,σ)=−1.
Note that if (W,σ) is a minimal WSP-pair, then sgn(c
χ
) = +1 for a transposition χ
and sgn(c
χ
)=(−1)
λ+1
if χ is of the form in Equation (4). In particular, when (W,σ)
is minimal and simple, its sign and weight is consistent with the definition given in the
introduction.
the electronic journal of combinatorics 13 (2006), #R37 10
Figure 6: A self-intersecting cycle and its corresponding pair of intersecting cycles
4 Proof of the Hamburger Theorem
As mentioned in Section 2, we prove Lemmas 2.1 and 2.2 for the weighted version of the
Hamburger Theorem, thereby proving Theorem 2.3. Theorem 1.1 follows as a special case
of Theorem 2.3.
4.1 Proof of Lemma 2.1, Part I
Recall Properties 1 and 2 as well as Lemma 2.1.
Property 1. The walk system contains a walk that has a self-intersection.
Property 2. The walk system has two intersecting walks, neither of which is a 2-cycle.
Lemma 2.1. The set of all walk systems W that satisfy either Property 1 or Property
2 can be partitioned into equivalence classes, each of which contributes a net zero to the
permutation expansion of the determinant of M
H
.
The proof of Lemma 2.1 is a generalization of the involution principle, the idea of which
comes from the picture presented in Figure 6. Given a self-intersecting walk, changing
the order of edge traversal at the vertex of self-intersection leads to breaking the one

self-intersecting walk into two walks that intersect at that same vertex. Since the edge
set of the collection of walks has not changed, the weight of the two WSP-pairs is the
same. We have introduced a transposition into the sign of the permutation cycle of the
WSP-pair; this changes the sign of the WSP-pair, so these two WSP-pairs will cancel in
the permutation expansion of the determinant of M
H
.
One can imagine that this means that to every self-intersecting WSP-pair we can asso-
ciate one WSP-pair with two cycles intersecting. However, more than one self-intersection
may occur at this same point, and there may be additional walks that pass through that
same point. Exactly what this WSP-pair would cancel with is not clear. If we decide
to break all the self-intersections so that we have some number N of walks through our
vertex, it is not clear how we should sew the cycles back together. One starts to get the
idea that we must consider all possible ways of sewing back together. Once we do just
that, we have a family F of WSP-pairs, all of the same weight, whose net contribution to
the permutation expansion of the determinant is zero.
This idea is conceptually simple but the proof is notationally complicated.
the electronic journal of combinatorics 13 (2006), #R37 11
If W satisfies either Property 1 or Property 2, then there is some vertex of intersection,
be it either a self-intersection or an intersection of two walks. Our aim is to choose a well-
defined first point of intersection at which we will build the family F. The initial term
order gives an order on walks associated to permutation cycles; we choose the earliest
walk c
χ
α
that has some vertex of intersection. Once we have determined the earliest walk,
we start at v
Φ(c
χ
α

)
and follow the walk
c
χ
a
: P
1
→ Q
1
→ P
2
→···→P
m
→ Q
m
,
until we reach a vertex of intersection.
In our discussion, we make the assumption that this first vertex of intersection is a
vertex v
∗
in G
1
. A similar argument exists if the first appearance occurs in G
2
.Notice
that at v
∗
there may be multiple self-intersections or multiple intersections of walks. We
will create a family F of WSP-pairs that takes into account each of these possibilities.
If we want to rigorously define the breaking of a self-intersecting walk at a vertex of

self-intersection, we need to specify many different components of the WSP-pair (C, σ).
First, we need to specify on which walk in W we are acting. Next, we need to specify the
vertex of self-intersection. Since this self-intersection vertex may occur in multiple paths,
we need to specify which two paths we interchange in the breaking process.
4.2 Definitions of Breaking and Sewing
In the following paragraphs, we define “breaking” on WSP-pairs, which takes in a WSP-
pair (W,σ), one of σ’s permutation cycles χ
α
, the associated walk c
χ
α
,pathsP
y
and P
z
in
c
χ
α
, and the vertex v
∗
in both P
y
and P
z
where c
χ
α
has a self-intersection. For simplicity,
we assume that v

∗
is not a distinguished vertex, but the argument still holds in that case.
The inverse of this operation is “sewing”.
In this framework, c
χ
α
has the form
c
χ
α
: P
1
→ Q
1
→ P
2
→···→P
y
→ Q
y
→···→P
z
→ Q
z
→···→P
l
→ Q
l
,
where the paths P

y
and P
z
in G
1
are separated into two halves as
P
y
: P
(1)
y
→ v
∗
→ P
(2)
y
and
P
z
: P
(1)
z
→ v
∗
→ P
(2)
z
.
Remember that P
y

and P
z
are paths that stop over at various vertices depending on
the permutation χ
α
. The vertex v
∗
must have adjacent stop-over vertices in each of the
two paths P
y
and P
z
. Let the nearest stop-over vertices on the paths P
y
and P
z
be v
ϕ
y
and v
ϕ
y+1
,andv
ϕ
z
and v
ϕ
z+1
, respectively.
This implies χ

α
has the form
χ
α
=(ϕ
11
··· ϕ
1µ
1
ω
11
··· ϕ
y
ϕ
y+1
··· ϕ
z
ϕ
z+1
··· ϕ
λµ
λ
ω
λ1
··· ω
λν
λ
).
the electronic journal of combinatorics 13 (2006), #R37 12
We can now precisely define the result of breaking. We define χ

β
and χ
γ
by splitting
χ
α
as follows:
χ
β
=(ϕ
11
··· ϕ
y
ϕ
z+1
··· ω
λν
λ
)
and
χ
γ
=(ϕ
y+1
··· ϕ
z
),
with the necessary rewriting of χ
γ
to have as its initial entry the value Φ(χ

γ
). Define c
χ
β
and c
χ
γ
to be
c
χ
β
: P
1
→ Q
1
→ P
2
→···→P
(1)
y
→ v
∗
→ P
(2)
z
→ Q
z
→···→P
l
→ Q

l
,
and
c
χ
γ
: P
(1)
z
→ v
∗
→ P
(2)
y
→ Q
y
→···→Q
z−1
,
again changing the starting vertex of c
χ
γ
to v
Φ(c
χ
γ
)
.
We define the breaking of the WSP-pair with the above inputs to be the WSP-pair
(W


,σ

) such that
W

= W∪{c
χ
β
,c
χ
γ
}\{c
χ
α
}
and
σ

= σχ
−1
α
χ
β
χ
γ
= σ · (ϕ
y+1
ϕ
z+1

).
TheedgesetofW is equal to the edge set of W

, so the weight of the modified cycle
systems is the same as the original. Since we changed σ to σ

by multiplying only by a
transposition, the sign of the modified WSP-pair is opposite to that of the original.
By only discussing the case when v
∗
is not distinguished, we avoid notational issues
brought upon by cases when v
∗
is or is not one of the stop-over vertices.
4.3 Proof of Lemma 2.1, Part II
Having defined breaking and sewing, we can continue the proof.
For any WSP-pair (W,σ) satisfying either Property 1 or Property 2, let c
χ
α
∈Wbe
the first walk in the initial term order with an vertex of intersection. Let v
∗
be the first
vertex of intersection in c
χ
α
. Then for all walks c with one or more self-intersections at v
∗
,
continue to break c at v

∗
until there are no more self-intersections. Define the resulting
WSP-pair (W
u
,σ
u
)tobetheunlinked WSP-pair associated to (W,σ). In (W
u
,σ
u
), there
is some number N of general walks intersecting at vertex v
∗
. There may be a 2-cycle
intersecting v
∗
as well, but this does not matter.
For any permutation ξ ∈ S
N
,letξ = ζ
1
ζ
2
···ζ
η
be its cycle representation, where each
ζ
ι
is a cycle. For each 1 ≤ ι ≤ η, sew together walks in order: if ζ
ι

=(δ
ι1
··· δ
ιε
ι
), sew
together c
χ
δ
ι1
and c
χ
δ
ι2
at v
∗
. Sew this result together with c
χ
δ
ι3
, and so on through c
χ
δ
ιε
ι
.
Note that the result of these sewings is unique, and that every WSP-pair (W,σ)with
(W
u
,σ

u
) as its unlinked WSP-pair can be obtained in this way, and no other WSP-pair
appears. We can perform this procedure for any ξ ∈ S
N
; the sign of the resulting walk
the electronic journal of combinatorics 13 (2006), #R37 13
system (C
ξ
,σ
ξ
)issgn(C
ξ
,σ
ξ
)=sgn(ξ)sgn(W
u
,σ
u
). This means that the contribution to
the determinant of the weights of all WSP-pairs in the family F is

ξ∈S
N
sgn(ξ)sgn(W
u
,σ
u
)wt(W
u
,σ

u
)=sgn(W
u
,σ
u
)wt(W
u
,σ
u
)

ξ∈S
N
sgn(ξ)=0. (6)
So elements of the same family cancel out in the determinant of M
H
, giving that the
contributions of all WSP-pairs satisfying either Property 1 or Property 2 cancel out in
the hamburger determinant. •
4.4 Proof of Lemma 2.2
Recall Properties 3 and 4 as well as Lemma 2.2.
Property 3. The walk system has two intersecting walks, one of which is a 2-cycle.
Property 4. The walk system is not minimal.
Lemma 2.2. The set of all walk systems W that satisfy neither Property 1 nor Property
2 and that satisfy Property 3 or Property 4 can be partitioned into equivalence classes,
each of which contributes a net zero to the permutation expansion of the determinant of
M
H
.
In the proof of Lemma 2.2, we do not base our family F around a singular vertex;

instead, we find a set of violations that each member of the family has. If the WSP-pair
satisfies the hypotheses of Lemma 2.2, then either there is some 2-cycle c : v
ϕ
→ w
ω
→ v
ϕ
that intersects with some other walk or there is some non-minimal permutation cycle.
Define a set of indices I ⊆ [k] of violations, of which an integer can become a member in
one of two ways. If (W,σ) is not minimal, there is at least one permutation cycle χ
α
with
more than two consecutive ϕ’s or ω’s in its cycle notation. For any intermediary ι between
two ϕ’s or ι + k between two ω’s, place ι in I. For example, if χ
α
=(··· ϕ

ιϕ

···),
we place ι ∈ I. Alternatively, there may be a 2-cycle c : v
i
→ w
k+i
→ v
i
such that either
v
i
is a vertex in some other walk c

χ
β
or w
k+i
is a vertex in some other cycle c
χ
γ
,orboth.
We declare this i to be in I as well.
Note that any WSP-pair (W,σ) satisfying either Property 3 or Property 4 has a non-
empty set I. From our original WSP-pair, obtain a minimal WSP-pair (W
m
,σ
m
)by
removing any transposition χ
α
from σ and its corresponding 2-cycle c
χ
α
from W,and
also for any non-minimal permutation cycle χ
β
we remove any intermediary ϕ’s or ω’s
from σ. We do not change the associated walk c
χ
β
in W since it still corresponds to this
minimized permutation cycle.
Let i be any element in I.Sincei ∈ I, the 2-cycle c

i
: v
i
→ w
k+i
→ v
i
intersects some
walk of W
m
either at v
i
,atw
k+i
, or both. So there are four cases:
Case 1. c
i
intersects a walk c
χ
β
at v
i
and no walk at w
k+i
.
Case 2. c
i
intersects a walk c
χ
γ

at w
k+i
and no walk at v
i
.
Case 3. c
i
intersects a walk c
χ
β
at v
i
and the same walk again at w
k+i
.
the electronic journal of combinatorics 13 (2006), #R37 14
3)
1) 2)
4)
Figure 7: The four cases in which a walk can intersect with a 2-cycle
Case 4. c
i
intersects a walk c
χ
β
at v
i
and another walk c
χ
γ

at w
k+i
.
See Figure 7 for a picture.
In order to build the family F, we modify the minimal WSP-pair (W
m
,σ
m
)ateach
i with some choice of options, each giving a possible WSP-pair that has (W
m
,σ
m
)asits
minimal WSP-pair. We define a set of two or four options O
i
for each i.
In Case 1, there are two options. Let o
1
be the option of including the walk c
i
in
W
m
and its corresponding transposition χ
α
in σ
m
.Leto
2

be the option of including the
intermediary ϕ = i in χ
β
in the position where c
χ
β
passes through v
i
.
[Note that we could not apply both options at the same time since the resulting χ
α
and χ
β
would not be disjoint and therefore is not a term in the permutation expansion of
the determinant.]
Similarly in Case 2, there are two options. Let o
1
be the option of including the walk
c
i
in W
m
and its corresponding transposition χ
α
in σ
m
.Leto
2
be the option of including
the intermediary ω = i + k in χ

γ
in the position where c
χ
γ
passes through w
k+i
.
In Case 3, there are four options. Let o
1
be the option of including the walk c
i
in
W
m
and its corresponding transposition χ
α
in σ
m
.Leto
2
be the option of including the
intermediary ϕ = i in χ
β
in the position where c
χ
β
passes through v
i
.Leto
3

be the option
of including the intermediary ω = i + k in χ
β
in the position where c
χ
β
passes through
w
k+i
.Leto
4
be the option of including both intermediaries ϕ = i and ω = i + k in χ
β
in
the respective positions where c
χ
β
passes through v
i
and w
k+i
.
In Case 4, there are four options. Let o
1
be the option of including the walk c
i
in
W
m
and its corresponding transposition χ

α
in σ
m
.Leto
2
be the option of including the
intermediary ϕ = i in χ
β
in the position where c
χ
β
passes through v
i
.Leto
3
be the option
of including the intermediary ω = i + k in χ
γ
in the position where c
χ
γ
passes through
w
k+i
.Leto
4
be the option of including intermediary ϕ = i in χ
β
in the position where
the electronic journal of combinatorics 13 (2006), #R37 15

sign = +1
(1364)(25) (12364)
sign = −1
Figure 8: A family of two walk systems that cancel in the hamburger determinant
c
χ
β
passes through v
i
and intermediary ω = i + k in χ
γ
in the position where c
χ
γ
passes
through w
k+i
.
Corresponding to the associated minimal WSP-pair (W
m
,σ
m
) and index set I, define
the family F to be the set of WSP-pairs (W
f
,σ
f
) by exercising all combinations of options
o
j

i
∈O
i
on (W
m
,σ
m
) for each i ∈ I. Note that every WSP-pair derived in this fashion is a
WSP-pair satisfying the hypotheses of Lemma 2.2 and is such that its associated minimal
WSP-pair is (W
m
,σ
m
). There is also no other WSP-pair (W

,σ

)with(W
m
,σ
m
)asits
minimal WSP-pair. See Figure 8 for a canceling family of two walk systems, one of which
is the non-minimal walk system from Figure 5.
Every WSP-pair in F has the same weight since each option changes the edge set
of W by at most a 2-cycle v
i
→ w
k+i
→ v

i
,where2≤ i ≤ k − 1 and each of those
2-cycles contributes a multiplier of 1 to the weight of the WSP-pair. The peculiar bounds
in this restriction are due to the structure of the hamburger graph. A walk through the
vertices v
1
and w
k+1
(or v
k
and w
2k
)mustuseedgee

1
(or e
k
), and include in its associated
permutation cycle 1 and k +1(ork and 2k). This implies that we can not have multiple
walks through any of these four special vertices, as that would yield two non-disjoint
permutation cycles in σ.
The sign of every WSP-pair in F is determined by the set of options applied to
(W
m
,σ
m
). Each option o
j
contributes a multiplicative ±1 depending on j—sgn(o
1

)=+1,
sgn(o
2
)=−1, and sgn(o
3
)=−1 and sgn(o
4
)=+1iftheyexist.
Each family F contributes a cumulative weight of 0. This is because there are the same
number of positive (W
f
,σ
f
) ∈Fas negative (W
f
,σ
f
) ∈ F .Weseethissinceifi
∗
∈ I then
any set of WSP-pairs that exercise the same options o
j
i
for all i = i
∗
has either two or four
members (depending on the case of i
∗
), which split evenly between positive and negative
WSP-pairs. Since each family contributes a net zero to the hamburger determinant, the

total contribution from WSP-pairs satisfying the hypotheses of Lemma 2.2 is zero. •
Since every WSP-pair appears once in the permutation expansion of det M
H
,theonly
WSP-pairs that contribute to the sum are those that are simple and minimal. Therefore
det M
H
is the sum over such WSP-pairs of sgn(W,σ)·wt(W,σ). This establishes Theorem
2.3 and gives Theorem 1.1 as a special case.
the electronic journal of combinatorics 13 (2006), #R37 16
Figure 9: The symmetric difference of two matchings gives a cycle system
5 Applications of the Hamburger Theorem
We discuss first the application of the Hamburger Theorem in the case where the region is
an Aztec diamond, mirroring results of Brualdi and Kirkland. Then we discuss the results
from the case where the region is an Aztec pillow, and along the way, we explain how to
implement the Hamburger Theorem when our region is any generalized Aztec pillow.
5.1 Domino Tilings and Digraphs
The Hamburger Theorem applies to counting the number of domino tilings of Aztec
diamonds and generalized Aztec pillows. To illustrate this connection, we count domino
tilings of the Aztec diamond by counting an equivalent quantity, the number of matchings
on the dual graph G of the Aztec diamond. We use the natural matching N of horizontal
neighbors in G, as exemplified in Figure 9a on AD
4
, as a reference point. Given any
other matching M on G, such as in Figure 9b, its symmetric difference with the natural
matching is a collection of cycles in the graph, such as in Figure 9c.
When we orient the edges in N from black vertices to white vertices and in M from
white vertices to black vertices, the symmetric difference becomes a collection of directed
cycles. Notice that edges in the upper half of G all go from left to right and the edges in
the bottom half of G go from right to left.

Since the edges of N always appear in the cycles, we can contract the edges of N
to vertices of a new graph which retains its structure in terms of the cycle systems it
produces. This new graph H is of the form in Figure 10a. This argument shows that the
number of domino tilings of an Aztec diamond equals the number of cycle systems of this
new condensed graph, called the region’s digraph.
We wish to concretize this notion of a digraph of the Aztec diamond AD
n
.Giventhe
natural tiling of an Aztec diamond consisting solely of horizontal dominoes, we place a
vertex in the center of every domino. The edges of this digraph fall into three families.
From every vertex in the top half of the diamond, create edges to the east, to the northeast,
and to the southeast whenever there is a vertex there. From every vertex in the bottom
half of the diamond, form edges to the west, to the southwest, and to the northwest
the electronic journal of combinatorics 13 (2006), #R37 17
Figure 10: The hamburger graph for an Aztec diamond and an Aztec pillow
whenever there is a vertex there. Additionally, label the bottom vertices in the top half
v
1
through v
n
from west to east and the top vertices in the bottom half w
n+1
through
w
2n
, also west to east. For all i between 1 and n, create a directed edge from v
i
to w
n+i
and from w

n+i
to v
i
. The result when this construction is applied to AD
4
is a graph of
the form in Figure 10a. By construction, this graph is a hamburger graph.
Theorem 5.1. The digraph of an Aztec diamond is a hamburger graph.
Since both the upper half of the digraph and the lower half of the digraph are strongly
planar, there are no negative cycle systems. This implies that the determinant of the cor-
responding hamburger matrix counts exactly the number of cycle systems in the digraph.
Corollary 5.2. The number of domino tilings of an Aztec diamond is the determinant of
its hamburger matrix.
5.2 Applying the Hamburger Theorem to Aztec Diamonds
In order to apply Theorem 1.1 to count the number of tilings of AD
n
, we need to find
the number of paths in the upper half of D from v
i
to v
j
and the number of paths in
the lower half of D from w
n+j
to w
n+i
. The key observation is that by the equivalence in
Figure 11, we are in effect counting the number of paths from (i, i)to(j, j) using steps of
size (0, 1), (1, 0), or (1, 1) that do not pass above the line y = x. Thisisacombinatorial
interpretation for the (j − i)-th large Schr¨oder number. The large Schr¨oder numbers

s
0
, ,s
5
are 1, 2, 6, 22, 90, 394. This is sequence number A006318 in the Encyclopedia of
Integer Sequences [13].
Corollary 5.3. The number of domino tilings of the Aztec diamond AD
n
is equal to
#AD
n
=det

S
n
I
n
−I
n
S
T
n

,
where S
n
is an upper triangular matrix with the (i, j)-th entry equal to s
j−i
for j ≥ i.
the electronic journal of combinatorics 13 (2006), #R37 18

Figure 11: The equivalence between paths in D and lattice paths in the first quadrant
For example, when n = 6, the matrix S
6
is
S
6
=








1 2 6 22 90 394
01262290
001 2 6 22
000 1 2 6
000 0 1 2
000 0 0 1








.

Brualdi and Kirkland prove a similar determinant formula for the number of tilings
of an Aztec diamond in a matrix-theoretical fashion based on the n(n +1)× n(n +1)
Kasteleyn matrix of the graph H and a Schur complement calculation. The Hamburger
Theorem gives a purely combinatorial way to reduce the calculation of the number of
tilings of an Aztec diamond to the calculation of a 2n × 2n Hamburger determinant.
Following cues from Brualdi and Kirkland, we can reduce this to an n× n determinant
via a Schur complement. For uses and the history of the Schur Complement, see the new
book [14] by Zhang. In the case of the block matrix M
H
in Equation (1), taking the Schur
complement of B in M
H
gives
det M
H
=det

AD
1
−D
2
B

· det

I 0
B
−1
D
2

I

=det

A + D
1
B
−1
D
2
D
1
0 B

=det(A + D
1
B
−1
D
2
) · det B
=det(A + D
1
B
−1
D
2
),
since B is a lower triangular matrix with 1’s on the diagonal. In this way, every hamburger
determinant can be reduced to a smaller determinant of a Schur complement matrix. In

the case of a simple hamburger graph where D
2
= D
1
= I, the determinant reduces
further to det(A + B
−1
). Lastly, in the case where the hamburger graph is rotationally
the electronic journal of combinatorics 13 (2006), #R37 19
symmetric, B = JAJ where J is the exchange matrix, which consists of 1’s down the
main skew-diagonal and 0’s elsewhere. This implies that we can write the determinant in
terms of just the submatrix A, i.e., det(A + JA
−1
J). (Note that J
−1
= J.)
Corollary 5.4. The number of domino tilings of the Aztec diamond AD
n
is equal to
det(S
n
+ J
n
S
−1
n
J
n
), where J
n

is the n × n exchange matrix.
In the case of a hamburger graph H, we call this Schur complement, A+B
−1
,areduced
hamburger matrix. In the Aztec diamond graph example above, we can thus calculate the
number of tilings of the Aztec diamond AD
6
as follows. The inverse of S
6
is
S
−1
6
=








1 −2 −2 −6 −22 −90
01−2 −2 −6 −22
001−2 −2 −6
0001−2 −2
0000 1−2
000001









,
which implies that the determinant of the reduced hamburger matrix
M
6
=








2 2 6 22 90 394
−2 2 2 6 22 90
−2 −222622
−6 −2 −222 6
−22 −6 −2 −22 2
−90 −22 −6 −2 −22









gives the number of tilings of AD
6
.
Brualdi and Kirkland were the first to find such a determinantal formula for the number
of tilings of an Aztec diamond [2]. Their matrix was different only in the fact that each
entry was multiplied by (−1) and there was a multiplicative factor of (−1)
n
. Brualdi and
Kirkland were able to calculate the sequence of determinants, {det M
n
},usingaJ-fraction
expansion, which only works when matrices are Toeplitz or Hankel.
Eu and Fu also found an n × n determinant that calculated the number of tilings of
an Aztec diamond [3]. Their matrix also involves large Schr¨oder numbers; when n =6,
their matrix is








12622 90 394
2 6 22 90 394 1806
6 22 90 394 1806 8558
22 90 394 1806 8558 41586

90 394 1806 8558 41586 206098
394 1806 8558 41586 206098 1037718








.
Their proof hinges on the relationship between domino tilings in an Aztec diamond and
path systems in a lattice derived from the Aztec diamond’s structure. The key observation
is that these path systems in the lattice can be extended uniquely outside the Aztec
the electronic journal of combinatorics 13 (2006), #R37 20
Figure 12: The digraph of a generalized Aztec pillow from the digraph of an Aztec diamond
diamond to form Schr¨oder paths. Applying the Gessel–Viennot method gives the above
matrix of Schr¨oder numbers. Unfortunately, Eu and Fu’s method does not generalize
further, as explained at the end of the next section.
5.3 Applying the Hamburger Theorem to Aztec Pillows
A generalized Aztec pillow can be produced by restricting the placement of certain domi-
noes in an Aztec diamond in order to generate the desired boundary. We define the
digraph of a generalized Aztec pillow to be the restriction of the digraph of an Aztec
diamond to the vertices on the interior of the pillow. An example is presented Figure
12. Since the generalized Aztec pillow’s digraph is a restriction of the Aztec diamond’s
digraph, we have the following corollary.
Corollary 5.5. The digraph of an Aztec pillow or a generalized Aztec pillow is a ham-
burger graph.
Aztec pillows were introduced in part because of an intriguing conjecture for the
number of their tilings given by Propp [12].

Conjecture 5.6 (Propp’s Conjecture). The number of tilings of an Aztec pillow AP
n
is a larger number squared times a smaller number. We write #AP
n
= l
2
n
s
n
. In addition,
depending on the parity of n, the smaller number s
n
satisfies a simple generating function.
For AP
2m
, the generating function is
∞

m=0
s
2m
x
m
=(1+3x + x
2
− x
3
)/(1 − 2x − 2x
2
− 2x

3
+ x
4
).
For AP
2m+1
, the generating function is
∞

m=0
s
2m+1
x
m
=(2+x +2x
2
− x
3
)/(1 − 2x − 2x
2
− 2x
3
+ x
4
).
the electronic journal of combinatorics 13 (2006), #R37 21
Figure 13: The equivalence between paths in D and lattice paths in the first quadrant
My hope for the Hamburger Theorem was that it would give a proof of Propp’s Conjecture.
We shall see that although we achieve a faster determinantal method to calculate #AP
n

,
the sequence of determinants is not calculable by a J-factor expansion, as was the case
in Brualdi and Kirkland’s work.
Using the same method as for Aztec diamonds, creating the hamburger graph H for
an Aztec pillow gives Figure 10b. Counting the number of paths from v
i
to v
j
and from
w
n+j
to w
n+i
in successively larger Aztec pillows gives us the infinite upper-triangular
array

S =(s
i,j
)ofmodified Schr¨oder numbers defined by the following combinatorial
interpretation. Let s
i,j
be the number of paths from (i, i)to(j, j) using steps of size
(0, 1), (1, 0), and (1, 1), not passing above the line y = x nor below the line y = x/2. The
equivalence between these paths and paths in the upper half of the hamburger graph is
shown in Figure 13. The principal 7 × 7 minor matrix

S
7
of


S is
the electronic journal of combinatorics 13 (2006), #R37 22

S
7
=










11251657224
01251657224
0012 621 82
0001 2 6 22
0000 1 2 6
0000 0 1 2
0000 0 0 1











.
This table of numbers is sequence number A114292 in the Encyclopedia of Integer Se-
quences [13].
Theorem 5.7. The number of domino tilings of an Aztec pillow of order n is equal to
#AP
n
=det


S
n
I
n
−I
n
J
n

S
n
J
n

,
where

S

n
is the n × n principal submatrix of

S and J
n
is the n × n exchange matrix.
As in the case of Aztec diamonds, we can calculate the reduced hamburger matrix
through a Schur calculation. The inverse of

S
6
is

S
−1
6
=








1 −10000
01−2 −1 −2 −5
001−2 −2 −5
0001−2 −2
00001−2

000001








and the resulting reduced hamburger matrix for AP
6
is
M
6
=








21251657
−22251657
−2 −222621
−5 −2 −2226
−5 −2 −1 −222
0000−12









.
This gives us a much faster way to calculate the number of domino tilings of an Aztec
pillow than was known previously. We have reduced the calculation of the O(n
2
) × O(n
2
)
Kasteleyn–Percus determinant to an n × n reduced hamburger matrix. To be fair, the
Kasteleyn–Percus matrix has −1, 0, and +1 entries while the reduced hamburger matrix
may have very large entries, which makes theoretical running time comparisons difficult.
Experimentally, when calculating the number of domino tilings of AP
14
using Maple 8.0
on a 447 MHz Pentium III processor, the determinant of the 112 × 112 Kasteleyn–Percus
matrix took 25.3 seconds while the determinant of the 14 × 14 reduced hamburger matrix
took less than 0.1 seconds.
the electronic journal of combinatorics 13 (2006), #R37 23
Figure 14: A simple generalized hamburger graph
Whereas we have a very understandable determinantal formula for the number of
tilings of the region, this does not translate into a proof of Propp’s Conjecture because
we cannot calculate the determinant of the matrices M
n
explicitly. We cannot apply a

J-fraction expansion as Brualdi and Kirkland did since the reduced hamburger matrix is
not Toeplitz or Hankel. Also, we might hope to apply Eu and Fu’s method. It is not hard
to construct the background lattice derived from generalized Aztec pillows; however, the
path systems that are associated to domino tilings of the region cannot be extended as
Eu and Fu did. A Gessel–Viennot-derived matrix still exists, but its structure does not
lend itself to a simple determinant calculation as in the case of Aztec diamonds.
6 Generalizations
In this section we present a counterexample to a natural generalization of the Hamburger
Theorem and discuss an intriguing generalization of Propp’s Conjecture.
6.1 Counterexample to Generalizations of the Hamburger The-
orem
The structure of the hamburger graph presented in Figure 1 seems restrictive, so the
question naturally arises whether it is somehow necessary. Can the edge set E
3
between
graphs G
1
and G
2
be an arbitrary bipartite graph? The answer in general is no.
Take for example the simple graph H in Figure 14. As one can count by hand, there
are 10 distinct cycle systems in H, as enumerated in Figure 15. Creating the hamburger
matrix that would correspond to this graph gives
M
H
=









1 1 1 100
0 1 1 010
0 0 1 001
−1 0 0 100
00−1110
0 −1 0 111








,
where the lower left block describes paths from w
4
, w
5
,andw
6
to v
1
, v
2
,andv

3
and we
have changed no other block of the matrix. This seems the most logical extension of the
hamburger matrix.
the electronic journal of combinatorics 13 (2006), #R37 24
Figure 15: The ten cycle systems for the generalized hamburger graph in Figure 14
The determinant of this matrix is −5. Even though one might expect to need a new
sign convention on cycle systems in generalized hamburger graphs, any sign convention
would necessarily conserve the parity of the number of cycle systems. Therefore there is no
+1/−1 labeling of the cycle systems in Figure 15 that would allow det M
H
=

c∈C
sgn(c).
This means that either the matrix for this generalized hamburger graph is not correct or
that the theorem does not hold in general.
The only hope for my idea of how to generalize the hamburger matrix is what happens
when one considers the difference between even and odd permutations. In the counterex-
ample above, the edge set E
3
sends vertex w
4
to vertex v
1
, w
5
to v
3
,andw

6
to v
2
.This
can be thought of as an odd permutation of a hamburger’s edge set (sending w
4
to v
1
, w
5
to v
2
,andw
6
to v
3
). In my limited calculations so far, whenever the permutation of the
natural edge set is even, the parity of the number of cycle systems, counted by hand, is
equal to the parity of the determinant of the matrix. I have not found a sign convention
for these graphs that explains the calculated determinant, but being equal in parity allows
for hope of a possible extension to the Hamburger Theorem.
6.2 Generalizing Propp’s Conjecture
A key motivation in the study of Aztec pillows is Propp’s Conjecture for 3-pillows, pre-
sented in Section 5.3. Through experimental calculations, I have expanded and extended
the conjecture.
For one, I extend the conjecture to odd pillows. For any odd number q,wecanuse
the notation AP
q
n
to represent the n-th q-pillow for odd q. Recall from the introduction

that this is a centrally symmetric region with steps of length q and central belt of size
2 × 2n. Calculating the number of tilings of AP
5
n
, AP
7
n
,andAP
9
n
for n at least up to
40 gives strong evidence that #AP
q
n
for q odd is always a larger number squared times a
smaller number. As with 3-pillows, we can write #AP
q
n
= l
2
n,q
s
n,q
,wheres
n,q
is relatively
the electronic journal of combinatorics 13 (2006), #R37 25